Preface |
Introduction |
Finite-Dimensional Sets / Part I: |
Lebesgue covering dimension / 1: |
Covering dimension / 1.1: |
The covering dimension of In / 1.2: |
Embedding sets with finite covering dimension / 1.3: |
Large and small inductive dimensions / 1.4: |
Exercises |
Hausdorff measure and Hausdorff dimension / 2: |
Hausdorff measure and Lebesgue measure / 2.1: |
Hausdorff dimension / 2.2: |
The Hausdorff dimension of products / 2.3: |
Hausdorff dimension and covering dimension / 2.4: |
Box-counting dimension / 3: |
The definition of the box-counting dimension / 3.1: |
Basic properties of the box-counting dimension / 3.2: |
Box-counting dimension of products / 3.3: |
Orthogonal sequences / 3.4: |
An embedding theorem for subsets of RN in terms of the upper box-counting dimension / 4: |
Prevalence, probe spaces, and a crucial inequality / 5: |
Prevalence / 5.1: |
Measures based on sequences of linear subspaces / 5.2: |
Embedding sets with dH (X - X) finite / 6: |
No linear embedding is possible when dH (X) is finite / 6.1: |
No modulus of continuity is possible for L-1 / 6.2: |
Thickness exponents / 7: |
The thickness exponent / 7.1: |
Lipschitz deviation / 7.2: |
Dual thickness / 7.3: |
Embedding sets of finite box-counting dimension / Exercises: |
Embedding sets with Hölder continuous parametrisation / 8.1: |
Sharpness of the Hölder exponent / 8.2: |
Assouad dimension / 9: |
Homogeneous spaces and the Assouad dimension / 9.1: |
Assouad dimension and products / 9.2: |
Homogeneity is not sufficient for a bi-Lipschitz embedding / 9.3: |
Almost bi-Lipschitz embeddings / 9.5: |
Sharpness of the logarithmic exponent / 9.6: |
Consequences for embedding compact metric spaces / 9.7: |
Finite-Dimentional Attractors / Part II: |
Partial differential equations and nonlinear semigroups / 10: |
Nonlinear semigroups and attractors / 10.1: |
Sobolev spaces and fractional power spaces / 10.2: |
Abstract semilinear parabolic equations / 10.3: |
The two-dimensional Navier-Stokes equations / 10.4: |
Attracting sets in infinite-dimensional systems / 11: |
Global attractors / 11.1: |
Existence of the global attractor / 11.2: |
Example 1: semilinear parabolic equations / 11.3: |
Example 2: the two-dimensional Navier-Stokes equations / 11.4: |
Bounding the box-counting dimension of attractors / 12: |
Coverings of T[B(0,1)] via finite-dimensional approximations / 12.1: |
A dimension bound when Df ∈ Lλ/2 (B), λ < 1/2 / 12.2: |
Finite dimension when Df ∈ L1 (X) / 12.3: |
Semilinear parabolic equations in Hilbert spaces / 12.4: |
Thickness exponents of attractors / 13: |
Zero thickness / 13.1: |
Zero Lipschitz deviation / 13.2: |
The Takens Time-Delay Embedding Theorem / 14: |
The finite-dimensional case / 14.1: |
Periodic orbits and the Lipschitz constant for ordinary differential equations / 14.2: |
The infinite-dimensional case / 14.3: |
Periodic orbits and the Lipschitz constant for semilinear parabolic equations / 14.4: |
Parametrisation of attractors via point values / 15: |
Real analytic functions and the order of vanishing / 15.1: |
Dimension and thickness of A in Cr (Ω, Rd) / 15.2: |
Proof of Theorem 15.1 / 15.3: |
Applications / 15.4: |
Solutions to exercises |
References |
Index |
Preface |
Introduction |
Finite-Dimensional Sets / Part I: |
Lebesgue covering dimension / 1: |
Covering dimension / 1.1: |
The covering dimension of In / 1.2: |